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To calculate the maximum energy of the swinger one could use either the gravitational potential energy at the swinger's highest position, $GPE=mg(h_{max}-h_{min})$, or the kinetic energy, $KE=mv^{2}/2$, at the lowest of the swinger's positions; where the speed (and therefore KE) would be greatest. Experimentally, this would involve measuring either the change in verticle height or the speed attained at the low-point, respectively.

The energy in the system is converted from one form to the other for every quarter-wavelength, e.g. high behind swing position to bottom of swing position. Equating the kinetic energy at the minimum height with the change in gravitational potential energy from the maximum height one obtains:

$mv^{2}/2=mg(h_{max}-h_{min})$.

This shows that the the maximum speed one can obtain is in fact indepenent of the mass of the person (it appears on both sides of the equation so it cancels out.)

Solving for $v$ we get: $v=\sqrt{2g(h_{max}-h_{min})}$.

For a normal garden-variety swing, the change in vertical height would be about $1m$. This would give a maximum speed of $v=\sqrt{2g(1)}=4.4ms^{-1}$. Where I used $g=9.81ms^{-2}$.

Bare in mind that this approach does not account for wind resistance which, IMO, would be pretty much negligable at these speeds anyway.

Can you explain how the results of the the GPE and KE equations are related? Do they result in the same answer? Also, I assume GPE and KE are measured in Joules?
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SebJun 13 '11 at 14:14

Hi @Seb, the GPE and KE are the two forms the energy could be in. The total energy available in the system is $E=GPE+KE$. When the swing is high it is changing direction, which implies the velocity is zero and therefore the KE is zero. All the total energy is therefore all in the form of the GPE at this moment. When the swing is low, the velocity is maximum, and therefore so is the KE. The GPE is necessarily zero at this point. All of E, GPE and KE are energies - so yes, they are all measured in Joules.
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qftmeJun 21 '11 at 8:51